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Does anyone know of a simple direct proof that if p is prime, then $\sqrt{p}$ is irrational?

I have always seen this proved by contradiction and have been trying unsuccessfully to prove it constructively. I searched this site and could not find the question answered without using contradiction.

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closed as off-topic by user223391, Davide Giraudo, user26857, 6005, KittyL Dec 30 '15 at 23:37

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    $\begingroup$ Since the definition of irrational is not rational, most proofs of irrationality are contradictions. $\endgroup$ – Michael Burr Dec 30 '15 at 22:48
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    $\begingroup$ Tim Gowers has some discussion here about how any such proof really must be by contradiction. $\endgroup$ – Casteels Dec 30 '15 at 22:50
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    $\begingroup$ Possible duplicate of Proving that for each prime number $p$, the number $\sqrt{p}$ is irrational $\endgroup$ – 6005 Dec 30 '15 at 23:18
  • $\begingroup$ OP--proofs not necessarly by contradiction can be found here, which the duplicate was closed as a duplicate of. $\endgroup$ – 6005 Dec 31 '15 at 22:39
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Maybe an elementary proof that I gave in a more general context, but I can't find it on the site, so I'll adapt it to this case.

Set $n=\lfloor \sqrt p\rfloor$. Suppose $\sqrt p$ is rational and let $m$ be the smallest positive integer such that $m\sqrt p$ is an integer. Consider $m'=m(\sqrt p-n)$; it is an integer, and $$ m'\sqrt p=m(\sqrt p-n)\sqrt p=mp-nm\sqrt p $$ is an integer too.

However, since $0\le \sqrt p-n <1$, we have $0\le m' <m $. Since $m$ is the smallest positive integer such that $ m\sqrt p$ is an integer, it implies $m'=0$, which means $\sqrt p=n$, hence $p=n^2$, which contradicts $p$ being prime.

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You could first prove that all square numbers have a prime decomposition with all primes of even powers.

For example if $k$ is square, and,

$k=\prod_{k=1}^{n}p_{k}^{e_{k}}$,

then each of $e_{k}$ must necessarily be even.

From there it is enough to show that primes have an odd power (namely 1) in their decomposition.

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For specific primes you can prove $x^2-p$ is irreducible over the rationals by proving it is irreducible modulo some other prime $q$. This means that $\sqrt{p}$ cannot be rational. This can probably be done in general using quadratic reciprocity.

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If you have the Rational Root Theorem available to you (https://en.wikipedia.org/wiki/Rational_root_theorem) you can apply it to $x^2-p$.

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